Moving Coframes: I. A Practical Algorithm

Acta Applicandae Mathematicae

Volumn 51, Issue 2, 1998, Pages 161-213

  Fels, Mark ^a   Olver, Peter J ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Computer vision; Differential invariant; Equivalence; Lie group; Lie pseudogroup; Moving frame; Symmetry

Indexed keywords

EID: 0001302031     PISSN: 01678019     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1005878210297     Document Type: Article

References (54)

1. Ackerman, M. and Hermann, R.: Sophus Lie's 1884 Differential Invariant Paper, Math. Sci. Press, Brookline, Mass., 1976.
2. Anderson, I. M.: Introduction to the variational bicomplex, Contemp. Math. 132 (1992), 51-73.
3. Bryant, R. L., Chern, S.-S., Gardner, R. B., Goldschmidt, H. L. and Griffiths, P. A.: Exterior Differential Systems, Math. Sci. Res. Inst. Publ. 18, Springer-Verlag, New York, 1991.
4. Buchberger, B.: Applications of Gröbner bases in nonlinear computational geometry, in: J. R. Rice (ed.), Mathematical Aspects of Scientific Software, IMA Vol. in Math. Appl. 14, Springer-Verlag, New York, 1988, pp. 59-87.
5. Calabi, E., Olver, P. J., Shakiban, C., Tannenbaum, A. and Haker, S.: Differential and numerically invariant signature curves applied to object recognition, Internat. J. Computer Vision, to appear.
6. Cartan, É.: La méthode du repère mobile, la théorie des groupes continus, et les espaces généralisés, Exposés de géométrie No. 5, Hermann, Paris, 1935.
7. Cartan, É.: Leçons sur la théorie des espaces à connexion projective, Cahiers scientifiques, 17, Gauthier-Villars, Paris, 1937.
8. Cartan, É.: La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile, Cahiers scientifiques 18, Gauthier-Villars, Paris, 1937.
9. Cartan, É.: Les systèmes différentiels extérieurs et leurs applications géométriques, Exposés de géométrie 14, Hermann, Paris, 1945.
10. Cartan, É.: Sur la structure des groupes infinis de transformations, in Oeuvres complètes, Part. II, Vol. 2, Gauthier-Villars, Paris, 1953, pp. 571-624.
11. Cartan, É.: Les problèmes d'équivalence, in Oeuvres complètes, Part. II, Vol. 2, Gauthier-Villars, Paris, 1953, pp. 1311-1334.
12. Chern, S.-S.: Moving frames, in Élie Cartan et les mathématiques d'aujourd'hui, Soc. Math. France, Astérisque, numéro hors série, 1985, pp. 67-77.
13. Faugeras, O.: Cartan's moving frame method and its application to the geometry and evolution of curves in the euclidean, affine and projective planes, in: J. L. Mundy, A. Zisserman, D. Forsyth (eds), Applications of Invariance in Computer Vision, Lecture Notes in Computer Sci. 825, Springer-Verlag, New York, 1994, pp. 11-46.
14. Fels, M. and Olver, P. J.: On relative invariants, Math. Ann. 308 (1997), 701-732.
15. Fels, M. and Olver, P. J.: Moving coframes, II. Regularization and theoretical foundations, Acta Appl. Math., to appear.
16. Gardner, R. B.: The Method of Equivalence and Its Applications, SIAM, Philadelphia, 1989.
17. Green, M. L.: The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces, Duke Math. J. 45 (1978), 735-779.
18. Griffiths, P. A.: On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41 (1974), 775-814.
19. Guggenheimer, H. W.: Differential Geometry, McGraw-Hill, New York, 1963.
20. Heredero, R. H. and Olver, P. J.: Classification of invariant wave equations, J. Math. Phys. 37 (1996), 6414-6438.
21. Hille, E.: Ordinary Differential Equations in the Complex Domain, Wiley, New York, 1976.
22. Hsu, L. and Kamran, N.: Classification of second-order ordinary differential equations admitting Lie groups of fiber-preserving symmetries, Proc. London Math. Soc. 58 (1989), 387-416.
23. Jensen, G. R.: Higher Order Contact of Submanifolds of Homogeneous Spaces, Lecture Notes in Math. 610, Springer-Verlag, New York, 1977.
24. Kamran, N.: Contributions to the study of the equivalence problem of Elie Cartan and its applications to partial and ordinary differential equations, Mém. Cl. Sci. Acad. Roy. Belg. 45 (1989), Fac. 7.
25. Kamran, N. and Olver, P. J.: Equivalence problems for first order Lagrangians on the line, J. Differential Equations 80 (1989), 32-78.
26. Kobayashi, S.: Canonical forms on frame bundles of higher order contact, Proc. Sympos. Pure Math. 3 (1961), 186-193.
27. Kumpera, A.: Invariants différentiels d'un pseudogroupe de Lie, J. Differential Geom. 10 (1975), 289-416.
28. Lie, S.: Über unendlichen kontinuierliche Gruppen, Christ. Forh. Aar. 8 (1883), 1-47; also Gesammelte Abhandlungen, Vol. 5, B. G. Teubner, Leipzig, 1924, pp. 314-360.
29. Lie, S.: Über unendlichen kontinuierliche Gruppen, Christ. Forh. Aar. 8 (1883), 1-47; also Gesammelte Abhandlungen, Vol. 5, B. G. Teubner, Leipzig, 1924, pp. 314-360.
30. Lie, S.: Über Differentialinvarianten, Math. Ann. 24 (1884), 537-578; also Gesammelte Abhandlungen, Vol. 6, B. G. Teubner, Leipzig, 1927, pp. 95-138; see [1] for an English translation.
31. Lie, S.: Über Differentialinvarianten, Math. Ann. 24 (1884), 537-578; also Gesammelte Abhandlungen, Vol. 6, B. G. Teubner, Leipzig, 1927, pp. 95-138; see [1] for an English translation.
32. Lie, S.: Die Grundlagen für die Theorie der unendlichen kontinuierlichen Transformationsgruppen, Leipzig. Ber. 43 (1891), 316-393; also Gesammelte Abhandlungen, Vol. 6, B. G. Teubner, Leipzig, 1927, pp. 300-364.
33. Lie, S.: Die Grundlagen für die Theorie der unendlichen kontinuierlichen Transformationsgruppen, Leipzig. Ber. 43 (1891), 316-393; also Gesammelte Abhandlungen, Vol. 6, B. G. Teubner, Leipzig, 1927, pp. 300-364.
34. Lie, S.: Zur allgemeinen Theorie der partiellen Differentialgleichungen beliebeger Ordnung, Leipz. Berich. 47 (1895), 53-128; also Gesammelte Abhandlungen, Vol. 4, B. G. Teubner, Leipzig, 1929, pp. 320-384.
35. Lie, S.: Zur allgemeinen Theorie der partiellen Differentialgleichungen beliebeger Ordnung, Leipz. Berich. 47 (1895), 53-128; also Gesammelte Abhandlungen, Vol. 4, B. G. Teubner, Leipzig, 1929, pp. 320-384.
36. Lie, S.: Gruppenregister, in Gesammelte Abhandlungen, Vol. 5, B. G. Teubner, Leipzig, 1924, pp. 767-773.
37. Lisle, I.: Equivalence transformations for classes of differential equations, PhD Thesis, University of British Columbia, Vancouver, 1992.
38. Medolaghi, P.: Classificazione delle equazioni alle derivate parziali del secondo ordine, ehe ammettono un gruppo infinito di trasformazioni puntuali, Ann. Mat. Pura Appl. 1(3) (1898), 229-263.
39. Moons, T., Pauwels, E., Van Gool, L. and Oosterlinck, A.: Foundations of semi-differential invariants, Internat. J. Comput. Vision 14 (1995), 25-48.
40. Olver, P. J.: Classical invariant theory and the equivalence problem for particle Lagrangians. I. Binary Forms, Adv. in Math. 80 (1990), 39-77.
41. Olver, P. J.: Applications of Lie Groups to Differential Equations, 2nd edn, Graduate Texts in Math. 107, Springer-Verlag, New York, 1993.
42. Olver, P. J.: Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, 1995.
43. Olver, P. J.: Pseudo-stabilization of prolonged group actions. I. The order zero case, Nonlinear Math. Phys. 4 (1997), 99-136.
44. Olver, P. J., Sapiro, G. and Tannenbaum, A.: Invariant geometric evolutions of surfaces and volumetric smoothing, SIAM J. Appl. Math. 57 (1997), 176-194.
45. Ovsiannikov, L. V.: Group Analysis of Differential Equations, Academic Press, New York, 1982.
46. Robart, T. and Kamran, N.: Sur la théorie locale des pseudogroupes de transformations continus infinis I, Math. Ann. 308 (1997), 593-613.
47. Singer, I. M. and Sternberg, S.: The infinite groups of Lie and Cartan. Part I (the transitive groups), J. Anal. Math. 15 (1965), 1-114.
48. Sternberg, S.: Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, N.J., 1964.
49. Tresse, A.: Sur les invariants différentiels des groupes continus de transformations, Acta Math. 18 (1894), 1-88.
50. Vessiot, E.: Sur l'intégration des systèmes différentiels qui admettent des groupes continues de transformations. Acta Math. 28 (1904), 307-349.
51. Weyl, H.: Cartan on groups and differential geometry, Bull. Amer. Math. Soc. 44 (1938), 598-601.
52. Weyl, H.: Classical Groups, Princeton Univ. Press, Princeton, N.J., 1946.
53. Wilczynski, E. J.: Protective Differential Geometry of Curves and Ruled Surfaces, B. G. Teubner, Leipzig, 1906.
54. Willmore, T. J.: Riemannian Geometry, Oxford University Press, Oxford, 1993.